The Stacks project

Lemma 21.13.2. Let $\mathcal{C}$ be a site. Let $K' \to K$ be a map of presheaves of sets on $\mathcal{C}$ whose sheafification is surjective. Set $K'_ p = K' \times _ K \ldots \times _ K K'$ ($p + 1$-factors). For every abelian sheaf $\mathcal{F}$ there is a spectral sequence with $E_1^{p, q} = H^ q(K'_ p, \mathcal{F})$ converging to $H^{p + q}(K, \mathcal{F})$.

Proof. Since sheafification is exact, we see that $(K_ p')^\# $ is equal to $(K')^\# \times _{K^\# } \ldots \times _{K^\# } (K')^\# $ ($p + 1$-factors). Thus we may replace $K$ and $K'$ by their sheafifications and assume $K \to K'$ is a surjective map of sheaves. After replacing $\mathcal{C}$ by a “larger” site as in Sites, Lemma 7.29.5 we may assume that $K, K'$ are objects of $\mathcal{C}$ and that $\mathcal{U} = \{ K' \to K\} $ is a covering. Then we have the Čech to cohomology spectral sequence of Lemma 21.10.6 whose $E_1$ page is as indicated in the statement of the lemma. $\square$

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